All Questions
Tagged with co.combinatorics pr.probability
802 questions
12
votes
2
answers
947
views
How rare are unholey permutations?
For $S\subset [n]:=\{1,2,\dotsc,n\}$, define $\delta(S)$ to be the number of $m\in S$ such that $m+1\notin S$.
Given a permutation $\pi$ of $[n]$, we define the holeyness $D(\pi)$ of $\pi$ as being $...
- 20.2k
4
votes
0
answers
216
views
How frequent are permutations with small interleaving?
For $S\subset [n]:=\{1,2,\dotsc,n\}$, define $\delta(S)$ to be the number of $m\in S$ such that $m+1\notin S$.
Let $\pi$ be a permutation of $[n]$. For simplicity, assume that $\pi$ is an $n$-cycle. ...
- 20.2k
3
votes
1
answer
109
views
Asymptotic estimate of the cardinality of $H_{2n}=\{\vec{a} \in [-N,N]^{2n}:\sum_{i=1}^{2n} a_i = 0\}$
Let's suppose $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{Z}$. We may then define:
\begin{equation}
S_n = \sum_{i=1}^n a_i \tag{1}
\end{equation}
Now, in order to estimate $\lvert ...
- 3,871
7
votes
0
answers
579
views
Guises of the noncrossing partitions (NCPs)
From "Noncrossing partitions in surprising locations" by Jon McCammond:
Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
- 10.5k
2
votes
0
answers
83
views
Zero-One law for Hamiltonian path subgraphs of Hamming Distance Graphs?
$(\alpha,\beta,d)$-Hamming Distance Graph $G_d(\alpha,\beta)$ for $\alpha,\beta\in(0,1]$ is a graph on $2^d$ vertices $v_0,\dots,v_{2^d-1}$ with edges $(v_i,v_j)\in\mathcal E(G_d)$ iff $0<\sum_{t=1}...
- 13.9k
2
votes
2
answers
256
views
Model for random graphs where clique number remains bounded
In the Erdös-Rényi model for random graphs,the clique number is seen to go to infinity as the number of vertices grows. Is anyone aware of models for random graphs with bounded clique number?
- 2,630
1
vote
1
answer
684
views
Probability that random Bernoulli matrix is full rank
This is probably known already, but I could not find a quick argument.
Let $M$ be an $n\times m$ binary matrix with iid Bernoulli$(1/2)$ entries, and $n>m$. Tikhomirov recently settled that the ...
- 1,096
6
votes
3
answers
563
views
Large deviations for discrete uniform distribution
(Not sure if this belongs on stack-exchange or overflow; let me know if I should switch it).
Given a sum of $n$ IID random variables $\{X_i\}_{i=1}^n$, each uniform on the integers $0,1,...,r$ for ...
- 435
-2
votes
2
answers
280
views
Balls into bins with random number of balls
In the classical balls into bins we throw $m$ balls into $n$ bins. We throw the balls independently of each other and the probability of choosing the bins is uniform. For $n=m$ it is known that the ...
8
votes
1
answer
598
views
Expected value of biggest distance of adjacent points uniformly picked in $[0,1]$
We pick $n\ge 2$ points in $[0,1]$ with uniform distribution. What is the expected value of the largest distance of $2$ adjacent points?
- 48.9k
3
votes
0
answers
94
views
Probability of a random collection of subsets being a cover
Consider the set $[n]=\{1,2,\ldots,n\}$. Suppose for each set $A\subseteq [n]$ I have a $p_A \in [0,1]$. I now create a random collection $\mathcal{W}\subseteq\mathcal{P}([n])$ of subsets of $[n]$ by ...
- 526
2
votes
0
answers
109
views
Average number of pieces of a random piecewise-linear function
Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
- 6,853
7
votes
0
answers
122
views
Discrepancy of the finite approximation of the Lebesgue measure
Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...
- 109k
3
votes
1
answer
341
views
Spectral radius of Markov averaging operator on graphs
The definition of Markov operator which I am familiar with:
For a graph $G=(V,E)$, Markov's operator upon a function
$\varphi:V\rightarrow\mathbb{C}$ , $\varphi\in L^2(G,\nu)$ ($\nu(v):=\deg(v)$) ...
- 33
13
votes
1
answer
564
views
Coincidences between average Catalan tableaux
There are Catalan number $C_n$ of standard Young tableaux of shape $(n,n)$, which we view as $2\times n$ matrices. Denote by $P_n$ the average of these matrices:
$$
P_n \, := \, \frac{1}{C_n} \, \...
- 17k
6
votes
1
answer
225
views
Restricted independent set of the cycle graph $C_{3n}$
Let $V$ be the vertices of the cycle graph $C_{3n}$. Suppose there is a partition of $V$ into sets of $3$, i.e. $V=\cup_{k=1}^{n}{V_k}$ where $|V_k|=3$ for $k$ in $1..n$.
QUESTION: Is it possible ...
- 9,501
2
votes
1
answer
142
views
The expectation of partition times needed separate two elements in a set
I met a problem which can be formulated as set partition.
Given a set $S=\{s_1,s_2,...,s_n\}$ having $n$ elements, I want to separate two elements, say $s_1,s_2$, in $S$ by repeatedly using set ...
- 23
4
votes
1
answer
264
views
Information for discovering an item-colour assignment in a combinatorial game
We are given a set $S=\{i_1, i_2, \ldots, i_n\}$ of items and a set $C=\{c_1, c_2, \ldots, c_m\}$ of colours. Each item in $S$ is tinted with one colour $c\in C$. Let $\mathcal{A}$ be the set of all ...
- 1,677
1
vote
2
answers
116
views
How to use probability to find a matching in a family of graphs?
In a conference, I heard that we can use some probabilistic methods to find a matching in some kind of graphs. I would like to see some examples of such technics. Can someone provide some references ...
- 57
4
votes
1
answer
839
views
A balls into bins problem with combinatorial constraints
We are given $m$ balls and $n$ bins, with $m \ge n$. Each bin can contain at most $c$ balls (we assume that $c$ is an even integer). In a sequential fashion, at each time step, one ball is placed into ...
- 1,677
1
vote
0
answers
117
views
Entropy of endpoints of a random walk in a dense graph
Let $p\in[0,1]$ be a constant and let $G$ be a graph with $n$ vertices and $\approx p\binom{n}{2}$ edges. If you'd like, consider $p=1/2$.
Let $X$ be a random vertex of $G$ chosen proportional to ...
- 761
1
vote
1
answer
134
views
Random optimization problem
Let $V$ be a set of $n$-dimensional vectors such that, for each ${\bf v}\in V$ and for each index $i\in [n-1]$, we have $0\le v_{i+1}\le v_i$. Let $P(\cdot)$ be a discrete probability distribution ...
- 1,677
2
votes
0
answers
58
views
Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$
Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try...
So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...
- 6,853
1
vote
1
answer
72
views
Independent identical distribution sequence
given a measurable function $\alpha: (\Omega, \mu) \to \mathbb{R}$, and transformation $\sigma : \Omega \to \Omega $.
I found an example such that $\alpha, \alpha\circ \sigma, \alpha\circ \sigma^2, \...
- 553
1
vote
1
answer
105
views
What is the distribution of a Cartesian power of a collection of iid uniform points? (renewed)
The following question was asked recently at https://mathoverflow.net/questions/326631/what-is-the-distribution-of-a-cartesian-power-of-a-collection-of-iid-uniform-poi :
Take a rectangle with ...
- 128k
3
votes
1
answer
108
views
Expected size of matchings in a cubic graph
Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$?
In other ...
- 9,501
1
vote
0
answers
46
views
Gilbert-Varshamov with weight condition
For $N$ integer, it is known (Gilbert-Varshamov) that there exists a subset $S$ of $\{0,1\}^N$ such that $|S|$ is still exponential in $N$ and such that two elements $x, x'$ in $S$ are at least ...
- 141
1
vote
0
answers
221
views
Different balls in bins: What is the probability distribution of the sum of the minimum of the two types of balls over all bins?
Assume that there are $N$ different bins and two different kinds of balls, $R$ red balls and $W$ white balls.
The red balls and the white balls are randomly distributed across the bins (that is, for ...
- 53
0
votes
0
answers
62
views
Probability of detecting small bias in a die in the low confidence regime
We are given a biased $m$-sided die: one of the sides has probability $\frac{1}{m} + \gamma$ and all the rest have probability $\frac{1}{m} - \frac{\gamma}{m-1}$ each. The goal is to figure out which ...
- 211
2
votes
1
answer
106
views
How to efficiently sample uniformly from the set of $p$-equipartitions of an $n$-set?
I have a question related to this one. For $n,p \in \mathbb{N}_+$ such that $p\mid n$, let $\mathcal{P}^{\rm eq}$ be the set of all equipartitions of $n$ in $p$ sets; i.e., in sets of equal size $\...
- 1,372
4
votes
0
answers
540
views
How should the proof of the XYZ theorem be understood?
The XYZ Theorem of Shepp [1] states the following for a given poset $P$. Consider the probability space of all the linear extensions of $P$, where each possible extension is equally likely. Then for ...
- 571
2
votes
2
answers
243
views
Birthday problem extension to unequal probabilities and multiple collisions
Let $p_1, ... ,p_k$ denote the probabilities of drawing bin $1, .. ,k$, where $\sum_{i = 1}^{k} p_i= 1$. My question is if we draw $n$ times, how can I show that the probability that all bins are ...
- 31
1
vote
1
answer
976
views
Concentration of the load of the maximally loaded bin ($m$ balls $n$ bins) with nonuniform bin probabilities
There is a common argument used when investigating the concentration of the maximally loaded bin (say $X$ is the maximum load) when $m$ balls are thrown into $n$ bins under the uniform distribution. I ...
- 10.4k
5
votes
0
answers
244
views
Distribution of point knowing target in optimal matching
I am a young PhD student in statistics.
Recently, papers (Ambrosio, Stra and Trevisan; Talagrand; Ledoux to cite but a few) tackled the problem of finding the expected cost in an optimal matching, ...
- 555
3
votes
1
answer
822
views
Open Problems in Random Graphs [closed]
I am a PhD student in mathematics. I'm interested in probabilistic methods in combinatorics and especially random graphs. I am looking for an open problem in this area for my PhD proposal. I know that ...
- 57
4
votes
0
answers
266
views
Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
- 24.9k
3
votes
1
answer
155
views
Expected size of the smallest preimage set
Let $f$ a function from $\{0, 1 \}^{2n}$ to $\{0, 1 \}^{n}$ uniformly picked at random. I would like to have an estimation of the expected size of the smallest premiage set of $f$, more formally $\...
- 215
2
votes
1
answer
179
views
How to uniformly sample a square (0,1)-matrix whose trace is 0 and whose row sums and column sums are the same?
Happy New Year!
Suppose I would like to sample a $n \times n$ (0,1)-matrix whose trace is 0, and whose row sums and column sums are all $m$ with $1 \le m \le n-1.$ How can I sample this matrix ...
- 131
8
votes
1
answer
171
views
On the existence of a particular type of finite measure on $\mathbb N$
Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: ...
- 1,209
10
votes
3
answers
2k
views
Mean maximum distance for N random points on a unit square
Following up on Mean minimum distance for N random points on a one-dimensional line and Mean minimum distance for N random points on a unit square (plane), I have the following questions.
Given N ...
- 193
0
votes
1
answer
144
views
A problem related to the comparison of two integer-valued random variables
Consider an urn containing red, blue and green balls (the situation is the same illustrated in this post).
Let $X$ be the non-negative, integer-valued random variable defined as the number of trials (...
- 145
3
votes
4
answers
1k
views
Apply doubly stochastic matrix M to a probability vector, then entropy increases?
Consider a vector $p =(p_1,\dots,p_n)$, $p_i>0$, $\sum p_i = 1$
and a matrix $M_{ij}$, which is doubly stochastic: $\sum_i M_{ij} = 1, \sum_j M_{ij} = 1, M_{ij} > 0$.
Question 1 Just apply ...
- 24.9k
1
vote
1
answer
338
views
Expected values of two non-negative, integer-valued random variables related to an urn problem
Consider an urn containing $c$ distinguishable balls, $\alpha$ of which are red, $\beta$ of which are blue, and $\gamma$ of which are green, and $\alpha+\beta+\gamma=c$. We assume $\alpha,\beta,\gamma&...
- 145
0
votes
1
answer
75
views
Expected sum of chosen coordinates in a random subset of a Hamming hypercube
Let $S$ = $\{v_1, v_2, ..., v_n\}$ denote a random subset of a Hamming hypercube of dimension $d$, where $n = |S|$ and $n \leq 2^d$. If $v_i$ = $\langle x^i_1x^i_2... x^i_d\rangle$ for all $i \in [1,n]...
- 11
6
votes
2
answers
283
views
Nonlinear boolean functions
Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...
- 333
7
votes
1
answer
257
views
Collecting proofs of the birth of the giant component
I want to collect different proofs of Erdös-Rényi result on the double jump of the largest connected component on $G(n,p)$ (or in $G(n,M)$.
I know the original proof of Erdös-Rényi, the proof that ...
- 1,561
5
votes
1
answer
412
views
Random pairs of commuting permutations
Let $\Omega_n \subseteq \mathrm{Sym}(n)^4$ be the set of all $4$-tuples $(\sigma_1,\sigma_2,\tau_1,\tau_2)$ of permutations of $\{1,\ldots,n\}$ such that $\sigma_j \tau_k = \tau_k \sigma_j$ for each ...
- 1,769
2
votes
2
answers
128
views
Spectral decomposition of a combinatorial matrix/Random walks on $s$-sets
$\newcommand{\Z}{\mathbb{Z}}
\newcommand{\J}{\mathcal{J}}
\newcommand{\la}{\lambda}
\newcommand{\1}{\mathbf{1}}
\newcommand{\R}{\mathbb{R}}$
Take any $n\in[3;\infty]$. Here and in what follows, $[k;\...
- 128k
3
votes
0
answers
330
views
Random walk on $\mathbb{R}$ with "sticky" origin
Let $P_i$, $N_i$, and $Z_i$, $i\in\mathbb{N}$ be r.v.'s with the $P_i$, $N_i$, and $Z_i$ being identically distributed with known pdf's $f_P$, $f_N$, and $f_Z$, respectively; and with no dependence ...
- 313
3
votes
1
answer
206
views
Component properties in Euclidean graphs with distance threshold
In the context of Euclidean graphs with vertices randomly embedded in either a 2D plane (for instance square with length $L$) or in 3D (similarly, cube of side $L$), where an edge between two given ...
- 649